Well-balanced finite volume schemes for nearly steady adiabatic flows

TL;DR

This paper introduces well-balanced finite volume schemes for Euler equations with gravitation, capable of preserving steady adiabatic flows including non-hydrostatic equilibria across various coordinate systems and equations of state.

Contribution

The paper presents a novel local steady state reconstruction method that enhances finite volume schemes to accurately handle nearly steady adiabatic flows in multiple geometries and with complex equations of state.

Findings

Schemes preserve steady adiabatic flows including non-hydrostatic equilibria.

Demonstrated superior performance over standard schemes in various numerical experiments.

Applicable to Cartesian, cylindrical, and spherical coordinates with flexible flux integration.

Abstract

We present well-balanced finite volume schemes designed to approximate the Euler equations with gravitation. They are based on a novel local steady state reconstruction. The schemes preserve a discrete equivalent of steady adiabatic flow, which includes non-hydrostatic equilibria. The proposed method works in Cartesian, cylindrical and spherical coordinates. The scheme is not tied to any specific numerical flux and can be combined with any consistent numerical flux for the Euler equations, which provides great flexibility and simplifies the integration into any standard finite volume algorithm. Furthermore, the schemes can cope with general convex equations of state, which is particularly important in astrophysical applications. Both first- and second-order accurate versions of the schemes and their extension to several space dimensions are presented. The superior performance of the well-balanced schemes compared to standard schemes is demonstrated in a variety of numerical experiments. The chosen numerical experiments include simple one-dimensional problems in both Cartesian and spherical geometry, as well as two-dimensional simulations of stellar accretion in cylindrical geometry with a complex multi-physics equation of state.